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1)Indicate the type of flow given by φ = √r cos(θ/2)

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1)Indicate the type of flow given by φ = √r cos(θ/2). Calculate and plot typical streamlines. 14. Consider a source of flow in 3D of strength S (dimensions L3/T), Then the flow is axisymmetric and independent of the θ-direction in the spherical coordinate system. Show by a mass balance that S = 4πr 2vr and hence derive Eq. (15.41) for the potential due to a unit source in 3D.

2)Verify the vector identity 2v = ( · v) −× ( × v) (15.83) Show that, for irrotational axisymmetric flows in cylindrical coordinates, the streamfunction satisfies the following equation: ∂2ψ ∂r2 − 1 r ∂2ψ ∂θ 2 + ∂2ψ ∂z2 = 0 Dimensional consistency. What are units for C1 in Eq. (15.10)? Confirm that the units agree on both sides.

3)Stokes flow past a cylinder. The case of flow past a cylinder of infinite length normal to the axis was also studied by Stokes. In view of the 2D nature of the problem, it is more convenient to work in r, θ coordinates. The governing equation is 4ψ = 0. What are the boundary conditions that can be imposed on ψ? Assume that the far-field approach velocity is a constant. In view of the far-field condition, the following solution looks reasonable: ψ = v∞ A r + Br + Crln r sin θ Try fitting the constants, and show that a uniformly valid solution cannot be obtained. (This is called the Stokes paradox.)

4)A rotating sphere. A sphere of radius R is rotating in an infinite fluid at an angular velocity of . Derive the following expression for the velocity field: vφ = r? sin θ R3 r3 Find an expression for the torque exerted by the sphere on the fluid. Answer: the torque is 8πμ?R3. Hint: assume a solution of the form vφ = f(r)sin θ, where f(r) is a function just of r to be determined from the φ-momentum balance.

5)The E4 operator in axisymmetric cylindrical coordinates. Consider the flow in cylindrical coordinates with no dependence on θ. How is the streamfunction defined? What is the non-vanishing component of vorticity. Show that the E2 operator takes the form E2 = ∂2 /∂r2 − 1 r ∂/ ∂r + ∂2 /∂z2 (15.82) and E4ψ = 0 for Stokes flow.

6)Properties of the Stokes equation. Verify that both the pressure and the vorticity field satisfy the Laplace equation for Stokes flow. Verify that the velocity field satisfies the biharmonic equation 4v = 0 (15.81) General solution for ψ. Derive the general solution for ψ given in the text (Eq. (15.9)).

7)The convection–diffusion problem. Solve the following problem: d2c dx2 − Pe dc dx = 0 Use the boundary conditions c(0) = 1 and c(1) = 0. Solve both for large Péclet number, Pe, where this is a singular perturbation problem, and for small Pe, where this is a regular perturbation problem. Compare your answers with the analytical and numerical solutions. Where is the boundary layer for large Pe? Is it at x = 0 or at x = 1?

8)Heat generation in a slab with variable thermal conductivity. Solve the problem of heat generation in a slab with a variable thermal conductivity. Show that the problem can be represented as dθ /dξ [(1 + βθ) dθ/ dξ] = −1 with the boundary condition of no flux at the center and convective heat loss at the surface, where β is the coefficient in the thermal conductivity relation. Thus β equal to zero is the base case of constant conductivity. Obtain a regular perturbation solution with β as the expansion parameter.

9)An agitated tank has an impeller diameter of 10 cm and operates at a speed of revolution of 10 r.p.s. The tank has a diameter of 20 cm and is filled up to a height of 20 cm with liquid. Find the following: (i) the power consumed assuming turbulent conditions, (ii) the length scale at which viscous dissipation takes place, and (iii) the corresponding velocity scale.

10)For gas–liquid dispersions in agitated vessels, the gas flow rate QG is also important. Show that an additional group, namely the flow number defined as QG/(d3 I) is needed in order to correlate the data. Hence the power number is then correlated as a function of the Reynolds number and the flow number. Search the literature and find the common correlations suggested for this.

11)Scaleup of agitated systems for equal mass transfer. The mass transfer coefficient in agitated gas–liquid systems is often proportional to the power per unit volume of the reactor rather than the total power P dissipated in the system. On this basic derive a scaleup criterion for equal-mass transfer for a small and a large reactor.

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